Conservative scales in packing problems

Abstract

Packing problems (sometimes also called cutting problems) are combinatorial optimization problems concerned with placement of objects (items) in one or several containers. Some packing problems are special cases of several other problems such as resource-constrained scheduling, capacitated vehicle routing, etc. In this paper we consider a bounding technique for one-and higher-dimensional orthogonal packing problems, called conservative scales (CS) (in the scheduling terminology, redundant resources). CS are related to the possible structure of resource consumption: filling of a bin, distribution of the resource to the jobs, etc. In terms of packing, CS are modified item sizes such that the set of feasible packings is not reduced. In fact, every CS represents a valid inequality for a certain binary knapsack polyhedron.CS correspond to dual variables of the set-partitioning model of a special 1D cutting-stock problem. Some CS can be constructed by (data-dependent) dualfeasible functions ((D)DFFs). We discuss the relation of CS to DFFs: CS assume that at most 1 copy of each object can appear in a combination, whereas DFFs allow several copies. The literature has investigated so-called extremal maximal DFFs (EMDFFs) which should provide very strong CS. Analogously, we introduce the notions of maximal CS (MCS) and extremal maximal CS (EMCS) and show that EMDFFs do not necessarily produce (E)MCS. We propose fast greedy methods to "maximize" a given CS. Using the fact that EMCS define facets of the binary knapsack polyhedron, we use lifted cover inequalities as EMCS. For higherdimensional orthogonal packing, we propose a Sequential LP (SLP) method over the set of CS and investigate its convergence. Numerical results are presented.

LP Bounds in an Interval-Graph Algorithm for Orthogonal-Packing Feasibility

“…Results for 3D instances. We do not know any previous OPP-3 instances in the literature, except those in [BKRS09,BKRS]. We generated smaller sets of instances by similar principles.…”

“…To prove infeasibility, we can employ relaxations of the problem. The paper [BKRS09] reviews several relaxations: volume bounds, conservative scales [FS04b,BKRS], 1D relaxations, and relaxations of ILP models.…”

“…This was done in [BKRS09] in a probing procedure. Moreover, the duals of the underlying set-covering LP provide efficient conservative scales [CM09,BKRS].…”

Constrained two‐dimensional guillotine cutting problem: upper‐bound review and categorization

“…It is a subproblem rising in the nodes of tree searches proposed to solve the nonguillotine case, or often in the context of the strip packing problem (Martello et al., ; Alvarez‐Valdés et al., ), hence the goal in using those upper bounds is different. Moreover, three types of upper bounds can be identified, two based on preemptive and nonpreemptive relaxations, recalling the idea of 1 D‐slice bounds, and one based on the conservative scales (CS; Fekete and Schepers, 2004), generalizing 1 D‐area bounds (details can be found in Belov et al., , ). For the bounds related to 1 D‐slice , no details will be given on the underlying ILP models.…”

Using dual feasible functions to construct fast lower bounds for routing and location problems

“…The explicit or implicit use of Dual Feasible Functions (DFFs) has a long tradition in optimizationsee, chronologically, the work in [15,19,28,11,5,12,4,2,9,6,26,3], or the survey [8]. The DFFs are most often used to generate: (i) valid inequalities in certain Integer Linear Programs (e.g., for the knapsack polytope) or (ii) high-quality fast lower bounds (LBs) for cutting and packing problems.…”